Let $X_1,...,X_n \sim Uniform(0,1)$. Harmonic mean is defined as:
$H_n = \frac {n}{\sum_{i=1}^n\frac{1}{X_i}}$
Find a.s. limit of this as $n \rightarrow \infty$
I already did the problem for both arithmetic and geometric means which relied on SLLN and CMP theorems. In the case of harmonic mean expectation of $1/X_i$ is infinite so I can't use SLLN. I feel like the limit should be 0 but I'm at a loss.
For every positive $t$ and every $n$, $H_n\lt H_n^{(t)}$ where $$ \frac1{H_n^{(t)}}=\frac1n\sum_{k=1}^n\frac1{X_n+t}. $$ By the SLLN, $H_n^{(t)}\to h_t$ almost surely when $n\to\infty$, where $$ \frac1{h_t}=E\left[\frac1{X_1+t}\right]. $$ Thus, for every positive $t$, almost surely, $$ \limsup_{n\to\infty}H_n\leqslant h_t. $$ Since $1/X_1$ is not integrable, $h_t\to0$ when $t\to0$, hence, almost surely, $$ \limsup_{n\to\infty}H_n\leqslant0. $$ Since every $H_n$ is nonnegative with full probability, this shows that $H_n\to0$ with full probability.